Jayalal Sarma

The rigidity of a matrix A for target rank r is the minimum number of entries of A that must be changed to ensure that the rank of the altered matrix is at most r. Since its introduction by Valiant (1977), rigidity and similar rank-robustness functions of matrices have found numerous applications in circuit complexity, communication complexity, and learning complexity. Almost all n × n matrices over an infinite field have a rigidity of (n − r)2. It is a long-standing open question to construct infinite families of explicit matrices even with superlinear rigidity when r = Ω(n).

For a graph G(V, E) (|V| = n) and a vertex s ∈ V, a weighting scheme (W : E ↔→ Z +) is called a min-unique (resp. max-unique) weighting scheme if, for any vertex v of the graph G, there is a unique path of minimum (resp. maximum) weight from s to v, where weight of a path is the sum of the weights assigned to the edges. Instead, if the number of paths of minimum (resp. maximum) weight is bounded by nc for some constant c, then the weighting scheme is called a min-poly (resp. max-poly) weighting scheme. In this article, we propose an unambiguous nondeterministic log-space (UL) algorithm for the problem of testing reachability graphs augmented with a min-poly weighting scheme. This improves the result in Reinhardt and Allender [2000], in which a UL algorithm was given for the case when the weighting scheme is min-unique. Our main technique involves triple inductive counting and generalizes the techniques of Immerman [1988], Szelepcśenyi [1988], and Reinhardt and Allender [2000], combined with a hashing technique due to Fredman et al. [1984] (also used in Garvin et al. [2014]).We combine this with a complementary unambiguous verification method to give the desired UL algorithm. At the other end of the spectrum, we propose a UL algorithm for testing reachability in layered DAGs augmented with max-poly weighting schemes. To achieve this, we first reduce reachability in layered DAGs to the longest path problem for DAGs with a unique source, such that the reduction also preserves the max-unique and max-poly properties of the graph. Using our techniques, we generalize the double inductive counting method in Limaye et al. [2009], in which the UL algorithm was given for the longest path problem on DAGs with a unique sink and augmented with a max-unique weighting scheme. An important consequence of our results is that, to show NL = UL, it suffices to design log-space computable min-poly (or max-poly) weighting schemes for layered DAGs.

The comparator circuit model was originally introduced by Mayr et al. (1992) (and further studied by Cook et al. (2014)) to capture problems that are not known to be P-complete but still not known to admit efficient parallel algorithms. The class CC is the complexity class of problems many-one logspace reducible to the Comparator Circuit Value Problem and we know that NLOG⊆CC⊆P. Cook et al. (2014) showed that CC is also the class of languages decided by polynomial size comparator circuit families. We study generalizations of the comparator circuit model that work over fixed finite bounded posets. We observe that there are universal comparator circuits even over arbitrary fixed finite bounded posets. Building on this, we show the following: • Comparator circuits of polynomial size over fixed finite distributive lattices characterize the class CC. When the circuit is restricted to be skew, they characterize LOG. Noting that (uniform) polynomial sized Boolean circuits (resp. skew) characterize P (resp. NLOG), this indicates a comparison between P vs CC and NLOG vs LOG problems.• Complementing this, we show that comparator circuits of polynomial size over arbitrary fixed finite lattices characterize the class P even when the comparator circuit is skew.• In addition, we show a characterization of the class NP by a family of polynomial sized comparator circuits over fixed finite bounded posets. As an aside, we consider generalizations of Boolean formulae over arbitrary lattices. We show that Spira's theorem (Spira, 1971) can be extended to this setting as well and show that polynomial sized Boolean formulae over finite fixed lattices capture the class NC1.These results generalize results in Cook et al. (2014) regarding the power of comparator circuits. Our techniques involve design of comparator circuits and finite posets. We then use known results from lattice theory to show that the posets that we obtain can be embedded into appropriate lattices. Our results give new methods to establish CC upper bounds for problems and also indicate potential new approaches towards the problems P vs CC and NLOG vs LOG using lattice theoretic methods.

We study projective dimension, a graph parameter, denoted by pd(G) for a bipartite graph G, introduced by Pudlák and Rödl (1992). For a Boolean function f (on n bits), Pudlák and Rödl associated a bipartite graph G f and showed that size of the optimal branching program computing f , denoted by bpsize(f ), is at least pd(G f ) (also denoted by pd(f )). Hence, proving lower bounds for pd(f ) implies lower bounds for bpsize(f ). Despite several attempts (Pudlák and Rödl (1992), Rónyai et al. (2000)), proving super-linear lower bounds for projective dimension of explicit families of graphs has remained elusive. We observe that there exist a Boolean function f for which the gap between the pd(f ) and bpsize(f )) is 2 Ω( n ) . Motivated by the argument in Pudlák and Rödl (1992), we define two variants of projective dimension: projective dimension with intersection dimension 1, denoted by upd(f ), and bitwise decomposable projective dimension, denoted by bitpdim(f ). We show the following results: (a) We observe that there exist a Boolean function f for which the gap between upd(f ) and bpsize(f ) is 2 Ω( n ) . In contrast, we also show that the bitwise decomposable projective dimension characterizes size of the branching program up to a polynomial factor. That is, there exists a constant c > 0 and for any function f , bitpdim(f )/6 ≤ bpsize(f ) ≤ (bitpdim(f )) c . (b) We introduce a new candidate family of functions f for showing super-polynomial lower bounds for bitpdim(f ). As our main result, for this family of functions, we demonstrate gaps between pd(f ) and the above two new measures for f : . We adapt Nechiporuk's techniques for our linear algebraic setting to prove the best-known bpsize lower bounds for bitpdim. Motivated by this linear algebraic setting of our main result, we derive exponential lower bounds for two restricted variants of pd(f ) and upd(f ) by observing that they are exactly equal to well-studied graph parameters-bipartite clique cover number and bipartite partition number, respectively.

A normal Hall subgroup N of a group G is a normal subgroup with its order coprime with its index. Schur-Zassenhaus theorem states that every normal Hall subgroup has a complement subgroup, that is a set of coset representatives H which also forms a subgroup of G. In this paper, we present a framework to test isomorphism of groups with at least one normal Hall subgroup, when groups are given as multiplication tables. To establish the framework, we first observe that a proof of Schur-Zassenhaus theorem is constructive, and formulate a necessary and sufficient condition for testing isomorphism in terms of the associated actions of the semidirect products, and isomorphisms of the normal parts and complement parts. We then focus on the case when the normal subgroup is abelian. Utilizing basic facts of representation theory of finite groups and a technique by Le Gall (STACS 2009), we first get an efficient isomorphism testing algorithm when the complement has bounded number of generators. For the case when the complement subgroup is elementary abelian, which does not necessarily have bounded number of generators, we obtain a polynomial time isomorphism testing algorithm by reducing to generalized code isomorphism problem, which asks whether two linear subspaces are the same up to permutation of coordinates. A solution to the latter can be obtained by a mild extension of the singly exponential (in the number of coordinates) time algorithm for code isomorphism problem developed recently by Babai et al. (SODA 2011). Enroute to obtaining the above reduction, we study the following computational problem in representation theory of finite groups: given two representations ? and ? of a group H over Z dp , p a prime, determine if there exists an automorphism φ : H → H, such that the induced representation ?φ = ? 0 φ and ? are equivalent, in time poly(

We study depth lower bounds against non-monotone circuits, parametrized by a new measure of non-monotonicity: the orientation of a function f is the characteristic vector of the minimum sized set of negated variables needed in any DeMorgan circuit (circuits where negations appear only at the leaves) computing f. We prove trade-off results between the depth and the weight/structure of the orientation vectors in any circuit C computing the CLIQUE function on an n vertex graph. We prove that if C is of depth d and each gate computes a Boolean function with orientation of weight at most w (in terms of the inputs to C), then d × w must be Ω(n). In particular, if the weights are o(n log k n), then C must be of depth ω(log k n). We prove a barrier for our general technique. However, using specific properties of the CLIQUE function (used in Amano Maruoka (2005)) and the Karchmer-Wigderson framework (Karchmer Wigderson (1988)), we go beyond the limitations and obtain lower bounds when the weight restrictions are less stringent. We then study the depth lower bounds when the structure of the orientation vector is restricted. Asymptotic improvements to our results (in the restricted setting) separates NP from NC. As our main tool, we generalize Karchmer-Wigderson games (Karchmer Wigderson (1988)) for monotone functions to work for non-monotone circuits parametrized by the weight/structure of the orientation. We also prove structural results about orientation and prove connections between number of negations and weight of orientations required to compute a function.

We contribute to the program of proving lower bounds on the size of branching programs solving the Tree Evaluation Problem introduced by Cook et al. [2012]. Proving a superpolynomial lower bound for the size of nondeterministic thrifty branching programs would be an important step toward separating NL from P using the tree evaluation problem. First, we show that Read-Once Nondeterministic Thrifty BPs are equivalent to whole black-white pebbling algorithms, thus showing a tight lower bound (ignoring polynomial factors) for this model. We then introduce a weaker restriction of nondeterministic thrifty branching programs called Bitwise Independence. The best known [Cook et al. 2012] nondeterministic thrifty branching programs (of size O(kh/2+)) for the tree evaluation problem are Bitwise Independent. As our main result, we show that any Bitwise Independent Nondeterministic Thrifty Branching Program solving BT2(h, k) must have at least (k/2)h/2 states. Prior to this work, lower bounds were known for nondeterministic thrifty branching programs only for fixed heights h = 2, 3, 4 [Cook et al. 2012]. We prove our results by associating a fractional blackwhite pebbling strategy with any bitwise independent nondeterministic thrifty branching program solving the Tree Evaluation Problem. Such a connection was not known previously, even for fixed heights. Our main technique is the entropy method introduced by Jukna and Zák [2001] originally in the context of proving lower bounds for read-once branching programs. We also show that the previous lower bounds known [Cook et al. 2012] for deterministic branching programs for the Tree Evaluation Problem can be obtained using this approach. Using this method, we also show tight lower bounds for any k-way deterministic branching program solving the Tree Evaluation Problem when the instances are restricted to have the same group operation in all internal nodes.

We initiate a complexity theoretic study of the language based graph reachability problem (L-REACH) : Fix a language L. Given a graph whose edges are labelled with alphabet symbols of the language L and two special vertices s and t, test if there is path P from s to t in the graph such that the concatenation of the symbols seen from s to t in the path P forms a string in the language L. We study variants of this problem with different graph classes and different language classes and obtain complexity theoretic characterizations for all of them. Our main results are the following: • Restricting the language using formal language theory we show that the complexity of L-REACH increases with the power of the formal language class. We show that there is a regular language for which the L-REACH is NL-complete even for undirected graphs. In the case of linear languages, the complexity of L-REACH does not go beyond the complexity of L itself. Further, there is a deterministic context-free language L for which L-DAGREACH is LogCFL-complete. • We use L-REACH as a lens to study structural complexity. In this direction we show that there is a language A in TC0 for which A-DAGREACH is NP-complete. Using this we show that P vs NP question is equivalent to P vs DAGREACH-1(P)1 question. This leads to the intriguing possibility that by proving DAGREACH-1(P) is contained in some subclass of P, we can prove an upward translation of separation of complexity classes. Note that we do not know a way to upward translate the separation of complexity classes.

Fix an algebraic structure (A,⁎). Given a graph G=(V,E) and the labelling function ϕ (ϕ:E→A) for the edges, two nodes s, t∈V, and a subset F⊆A, the A-REACH problem asks if there is a path p (need not be simple) from s to t whose yield (result of the operation in the ordered set of the labels of the edges constituting the path) is in F. On the complexity frontier of this problem, we show the following results. • When A is a group whose size is polynomially bounded in the size of the graph (hence equivalently presented as a multiplication table at the input), and the graph is undirected, the A-REACH problem is in L. Building on this, using a decomposition in [4], we show that, when A is a fixed quasi-group, and the graph is undirected, the A-REACH problem is in L. In a contrast, we show NL-hardness of the problem over bidirected graphs, when A is a matrix group over Q, or an aperiodic monoid. • As our main theorem, we prove a dichotomy for graphs labelled with fixed aperiodic monoids by showing that for every fixed aperiodic monoid A, A-REACH problem is either in L or is NL-complete. • We show that there exists a monoid M, such that the reachability problem in general DAGs can be reduced to A-REACH problem for planar non-bipartite DAGs labelled with M. In contrast, we show that if the planar DAGs that we obtain above are bipartite, the problem can be further reduced to reachability testing in planar DAGs and hence is in UL.

The well-known SENSITIVITY CONJECTURE regarding combinatorial complexity measures on Boolean functions states that for any Boolean function f:{0,1} n →{0,1}, block sensitivity of f is polynomially related to sensitivity of f (denoted by s(f)). From the complexity theory side, the XOR LOG-RANK CONJECTURE states that for any Boolean function, f:{0,1} n →{0,1} the communication complexity of a related function f ⊕ :{0,1} n ×{0,1} n →{0,1} (defined as f ⊕ (x,y)=f(x⊕y)) is bounded by polynomial in logarithm of the sparsity of f (the number of non-zero Fourier coefficients for f, denoted by sparsity(f)). Both the conjectures play a central role in the domains in which they are studied. A recent result of Lin and Zhang (2017) implies that to confirm the above two conjectures it suffices to upper bound alternation of f (denoted by alt(f)) for all Boolean functions f by polynomial in s(f) and logarithm of sparsity(f), respectively. In this context, we show the following results: • We show that there exists a family of Boolean functions for which alt(f) is at least exponential in s(f) and alt(f) is at least exponential in log⁡sparsity(f). En route to the proof, we also show an exponential gap between alt(f) and the decision tree complexity of f, which might be of independent interest. • As our main result, we show that, despite the above exponential gap between alt(f) and log⁡sparsity(f), the XOR LOG-RANK CONJECTURE is true for functions with the alternation upper bounded by poly(log⁡n). It is easy to observe that the SENSITIVITY CONJECTURE is also true for this class of functions. • The starting point for the above result is the observation (derived from Lin and Zhang (2017)) that for any Boolean function f, deg(f)≤alt(f)deg 2 (f)deg m (f) where deg(f), deg 2 (f) and deg m (f) are the degrees of f over R, F 2 and Z m respectively. We give three further applications of this bound: (1) We show that for Boolean functions f of constant alternation have deg 2 (f)=Ω(log⁡n). (2) Moreover, these functions also have high sparsity (exp⁡(Ω(deg(f))), thus partially answering a question of Kulkarni and Santha (2013). (3) We observe that our relation upper bounding real degree also improves the upper bound for influence to deg 2 (f) 2 ⋅alt(f) improving Guo and Komargodski (2017).